What conserved quantity does the signature flip at |a| = 2√q saturate?

Object. On a surface with h = f₁ + f₂ the intersection form flips signature — (1,3) ↔ (2,2) — exactly as |a| crosses 2√q, the Weil bound (measured in PN-001: the orthogonal complement's top eigenvalue −0.033 at a = 5, +0.101 at a = 6). CASE-08 uses this flip as evidence that positivity must be imported from a real structure. This case sets the surface aside and takes the crossing itself as the object: at |a| = 2√q something is saturated, and the question is what — a conserved quantity, a bound, an index — as a property of the crossing and not of the particular rank-4 lattice it was first seen in.

Presenting problem (the friction). The flip is currently stated as a fact about one surface's intersection form. But 2√q is the Weil bound, which is not a fact about that surface — it is a statement about the eigenvalues of Frobenius, true across the whole family. So the crossing should have a formulation that never mentions the surface. If it does not — if the signature genuinely needs the rank-4 lattice to be stated — then CASE-08's transfer of the flip to the adelic object is unlicensed, and its strongest piece of evidence is an artefact of the example.

Differential. 1. The crossing saturates a bound on Frobenius eigenvalues (the Weil bound itself) — a statement about the zeta function of the family, needing no surface; the signature is a shadow of it. 2. The crossing is genuinely a lattice-signature fact and does not survive removal of the rank-4 intersection form; its appearing to sit at "the Weil bound" is a coincidence of this example. 3. The crossing saturates an index — a difference of dimensions that changes sign — and the conserved quantity is that index, statable wherever a trace and a pairing exist, which would carry it to the adelic side.

Why it matters (cross-link to CASE-08). PN-001's House left CASE-08 with a fork: is the adele-class programme looking for a geometry (a space with an ample class) or for a bound (a threshold a parameter crosses)? If the crossing at 2√q is the saturation of a conserved quantity statable with no surface, CASE-08 is looking for a bound and its four geometric accounts are the wrong register. If no surface-free statement exists, CASE-08 is looking for a geometry after all. Resolving this case tells CASE-08 which.

Cross-refs. Parent case: CASE-08 (what geometry supplies the polarization). Panel: PN-001 (its BACKPROP opened this case).